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This was an exam question and I was absolutely lost.

Find positive constants $a,b$ such that the function

$$ \frac{1}{z^2+a} + \frac{1}{z+b} $$

has three Laurent series, one for each of the domains $|z|<1, \,\, 1<|z|<4, \,\, |z|>4$.

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Try $a=1$ and $b=4$ or $a=16$ and $b=1$.

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